
SCIENTIFIC AMERICAN SUPPLEMENT NO. 531
NEW YORK, MARCH 6, 1886
Scientific American Supplement. Vol. XXI, No. 531.
Scientific American established 1845
Scientific American Supplement, $5 a year.
Scientific American and Supplement, $7 a year.
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TABLE OF CONTENTS.
I. CHEMISTRY AND METALLURGY.—Annatto.Analyses of the same.—By WM. LAWSON
Aluminum.—By J.A. PRICE.—Iron the basis of civilization.— Aluminum the metal of the future.—Discovery of aluminum.—Art of obtaining the metal.—Uses and possibilities
II. ENGINEERING AND MECHANICS.—The Use of Iron in Fortification. —Armorplated casements.—The SchumannGruson chilled iron cupola.—Mougin's rolled iron cupola.—With full page of engravings
High Speed on the Ocean
Sibley College Lectures.—Principles and Methods of Balancing Forces developed in Moving Bodies.—Momentum and centrifugal force.—By CHAS.T. PORTER.—3 figures
Compressed Air Power Schemes.—By J. STURGEON.—Several figures
The Berthon Collapsible Canoe.—2 engravings
The Fiftieth Anniversary of the Opening of the First German Steam Railroad.—With full page engraving
Improved Coal Elevator.—With engraving
III. TECHNOLOGY.—Steelmaking Ladles.—4 figures
Water Gas.—The relative value of water gas and other gases as Ironreducing Agents.—By B.H. THWAITE.—Experiments.—With tables and 1 figure
Japanese Rice Wine and Soja Sauce.—Method of making
IV. ELECTRICITY, MICROSCOPY, ETC.Apparatus for demonstrating that Electricity develops only on the Surface of Conductors.—1 figure
The Colson Telephone.—3 engravings
The Meldometer.—An apparatus for determining the melting points of minerals
Touch Transmission by Electricity in the Education of Deaf Mutes.—By S. TEFFT WALKER.—With 1 figure
V. HORTICULTURE.—Candelabra Cactus and the California Woodpecker.—By C.F. HOLDER.—With 2 engravings
How Plants are reproduced.—By C.E. STUART.—A paper read before the Chemists' Assistants' Association
VI. MISCELLANEOUS—The Origin of Meteorites.—With 1 figure
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THE USE OF IRON IN FORTIFICATION.
Roumania is thinking of protecting a portion of the artillery of the forts surrounding her capital by metallic cupolas. But, before deciding upon the mode of constructing these formidable and costly affairs, and before ordering them, she has desired to ascertain their efficacy and the respective merits of the chilled iron armor which was recently in fashion and of rolled iron, which looks as if it were to be the fashion hereafter.
The Krupp works have recommended and constructed a cupola of casehardened iron, while the Saint Chamond works have offered a turret of rolled iron. Both of these recommend themselves by various merits, and by remarkably ingenious arrangements, and it only remains to be seen how they will behave under the fire of the largest pieces of artillery.
We are far in advance of the time when cannons with smooth bore were obliged to approach to within a very short range of a scarp in order to open a breach, and we are far beyond that first rifled artillery which effected so great a revolution in tactics.
Today we station the batteries that are to tear open a rampart at distances therefrom of from 1,000 to 2,000 yards, and the long, 6 inch cannon that arms them has for probable deviations, under a charge of 20 pounds of powder, and at a distance of 1,000 yards, 28 feet in range, 16 inches in direct fire and 8 inches in curved.
The weight of the projectile is 88 pounds, and its remanent velocity at the moment of impact is 1,295 feet. Under this enormous live force, the masonry gradually crumbles, and carries along the earth of the parapet, and opens a breach for the assaulting columns.
In order to protect the masonry of the scarp, engineers first lowered the cordon to the level of the covertway. Under these circumstances, the enemy, although he could no longer see it, reached it by a curved or "plunging" shot. When, in fact, for a given distance we load a gun with the heaviest charge that it will stand, the trajectory, AMB (Fig. 2), is as depressed as possible, and the angles, a and a', at the start and arrival are small, and we have a direct shot. If we raise the chase of the piece, the projectile will describe a curve in space which would be a perfect parabola were it not for the resistance of the air, and the summit of such curve will rise in proportion as the angle so increases. So long as the falling angle, a, remains less than 45 deg., we shall have a curved shot. When the angle exceeds this, the shot is called "vertical." If we preserve the same charge, the parabolic curve in rising will meet the horizontal plane at a greater distance off. This is, as well known, the process employed for reaching more and more distant objects.
The length of a gun depends upon the maximum charge burned in it, since the combustion must be complete when the projectile reaches the open air. It results from this that although guns of great length are capable of throwing projectiles with small charges, it is possible to use shorter pieces for this purpose—such as howitzers for curved shots and mortars for vertical ones. The curved shot finds one application in the opening of breaches in scarp walls, despite the existence of a covering of great thickness. If, from a point, a (Fig. 3), we wish to strike the point, b, of a scarp, over the crest, c, of the covertway, it will suffice to pass a parabolic curve through these three points—the unknown data of the problem, and the charge necessary, being ascertained, for any given piece, from the artillery tables. In such cases it is necessary to ascertain the velocity at the impact, since the force of penetration depends upon the live force (mv squared) of the projectile, and the latter will not penetrate masonry unless it have sufficient remanent velocity. Live force, however, is not the sole factor that intervenes, for it is indispensable to consider the angle at which the projectile strikes the wall. Modern guns, such as the Krupp 6 inch and De Bange 6 and 8 inch, make a breach, the two former at a falling angle of 22 deg., and the latter at one of 30 deg.. It is not easy to lower the scarps enough to protect them from these blows, even by narrowing the ditch in order to bring them near the covering mass of the glacis.
The same guns are employed for dismounting the defender's pieces, which he covers as much as possible behind the parapet. Heavy howitzers destroy the materiel, while shrapnel, falling nearly vertically, and bursting among the men, render all operations impossible upon an open terreplein.
The effect of 6 and 8 inch rifled mortars is remarkable. The Germans have a 9 inch one that weighs 3,850 pounds, and the projectile of which weighs 300. But French mortars in nowise cede to those of their neighbors; Col. De Bange, for example, has constructed a 101/2 inch one of wonderful power and accuracy.
Seeing the destructive power of these modern engines of war, it may well be asked how many pieces the defense will be able to preserve intact for the last period of a siege—for the very moment at which it has most need of a few guns to hold the assailants in check and destroy the assaulting columns. Engineers have proposed two methods of protecting these few indispensable pieces. The first of these consists in placing each gun under a masonry vault, which is covered with earth on all sides except the one that contains the embrasure, this side being covered with armor plate.
The second consists in placing one or two guns under a metallic cupola, the embrasures in which are as small as possible. The cannon, in a vertical aim, revolves around the center of an aperture which may be of very small dimensions. As regards direct aim, the carriages are absolutely fixed to the cupola, which itself revolves around a vertical axis. These cupolas may be struck in three different ways: (1) at right angles, by a direct shot, and consequently with a full charge—very dangerous blows, that necessitate a great thickness of the armor plate; (2) obliquely, when the projectile, if the normal component of its real velocity is not sufficient to make it penetrate, will be deflected without doing the plate much harm; and (3) by a vertical shot that may strike the armor plate with great accuracy.
General Brialmont says that the metal of the cupola should be able to withstand both penetration and breakage; but these two conditions unfortunately require opposite qualities. A metal of sufficient ductility to withstand breakage is easily penetrated, and, conversely, one that is hard and does not permit of penetration does not resist shocks well. Up to the present, casehardened iron (Gruson) has appeared to best satisfy the contradictory conditions of the problem. Upon the tempered exterior of this, projectiles of chilled iron and cast steel break upon striking, absorbing a part of their live force for their own breakage.
In 1875 Commandant Mougin performed some experiments with a chilled iron turret established after these plans. The thickness of the metal normally to the blows was 231/2 inches, and the projectiles were of cast steel. The trial consisted in firing two solid 12 in. navy projectiles, 46 cylindrical 6 in. ones, weighing 100 lb., and 129 solid, pointed ones, 12 in. in diameter. The 6 inch projectiles were fired from a distance of 3,280 feet, with a remanent velocity of 1,300 feet. The different phases of the experiment are shown in Figs. 4, 5, and 6. The cupola was broken; but it is to be remarked that a movable and wellcovered one would not have been placed under so disadvantageous circumstances as the one under consideration, upon which it was easy to superpose the blows. An endeavor was next made to substitute a tougher metal for casehardened iron, and steel was naturally thought of. But hammered steel broke likewise, and a mixed or compound metal was still less successful. It became necessary, therefore, to reject hard metals, and to have recourse to malleable ones; and the one selected was rolled iron. Armor plate composed of this latter has been submitted to several tests, which appear to show that a thickness of 18 inches will serve as a sufficient barrier to the shots of any gun that an enemy can conveniently bring into the field.
Armor Plated Casemates.—Fig. 7 shows the state of a chilled iron casemate after a vigorous firing. The system that we are about to describe is much better, and is due to Commandant Mougin.
The gun is placed under a vault whose generatrices are at right angles to the line of fire (Fig. 8), and which contains a niche that traverses the parapet. This niche is of concrete, and its walls in the vicinity of the embrasure are protected by thick iron plate. The rectangular armor plate of rolled iron rests against an elastic cushion of sand compactly rammed into an iron plate caisson. The conical embrasure traverses this cushion by means of a caststeel piece firmly bolted to the caisson, and applied to the armor through the intermedium of a leaden ring. Externally, the cheeks of the embrasure and the merlons consist of blocks of concrete held in caissons of strong iron plate. The surrounding earthwork is of sand. For closing the embrasure, Commandant Mougin provides the armor with a disk, c, of heavy rolled iron, which contains two symmetrical apertures. This disk is movable around a horizontal axis, and its lower part and its trunnions are protected by the sloping mass of concrete that covers the head of the casemate. A windlass and chain give the disk the motion that brings one of its apertures opposite the embrasure or that closes the latter. When this portion of the disk has suffered too much from the enemy's fire, a simple maneuver gives it a half revolution, and the second aperture is then made use of.
The SchumannGruson Chilled Iron Cupola.—This cupola (Fig. 9) is domeshaped, and thus offers but little surface to direct fire; but it can be struck by a vertical shot, and it may be inquired whether its top can withstand the shock of projectiles from a 10 inch rifled mortar. It is designed for two 6 inch guns placed parallel. Its internal diameter is 191/2 feet, and the dome is 8 inches in thickness and has a radius of 161/2 feet. It rests upon a pivot, p, around which it revolves through the intermedium of rollers placed in a circle, r. The dome is of relatively small bulk—a bad feature as regards resistance to shock. To obviate this difficulty, the inventor partitions it internally in such a way as to leave only sufficient space to maneuver the guns. The partitions consist of iron plate boxes filled with concrete. The form of the dome has one inconvenience, viz., the embrasure in it is necessarily very oblique, and offers quite an elongated ellipse to blows, and the edges of the bevel upon a portion of the circumference are not strong enough. In order to close the embrasure as tightly as possible, the gun is surrounded with a ring provided with trunnions that enter the sides of the embrasure. The motion of the piece necessary to aim it vertically is effected around this axis of rotation. The weight of the gun is balanced by a system of counterpoises and the chains, l, and the breech terminates in a hollow screw, f, and a nut, g, held between two directing sectors, h. The cupola is revolved by simply acting upon the rollers.
Mougin's Rolled Iron Cupola.—The general form of this cupola (Fig. 1) is that of a cylindrical turret. It is 123/4 feet in diameter, and rises 31/4 feet above the top of the glacis. It has an advantage over the one just described in possessing more internal space, without having so large a diameter; and, as the embrasures are at right angles with the sides, the plates are less weakened. The turret consists of three plates assembled by slit and tongue joints, and rests upon a ring of strong iron plate strengthened by angle irons. Vertical partitions under the cheeks of the gun carriages serve as cross braces, and are connected with each other upon the table of the hydraulic pivot around which the entire affair revolves. This pivot terminates in a plunger that enters a strong steel presscylinder embedded in the masonry of the lower concrete vault.
The iron plate ring carries wheels and rollers, through the intermedium of which the turret is revolved. The circular iron track over which these move is independent of the outer armor.
The whole is maneuvered through the action of one man upon the piston of a very small hydraulic press. The guns are mounted upon hydraulic carriages. The brake that limits the recoil consists of two bronze pump chambers, a and b (Fig. 10). The former of these is 4 inches in diameter, and its piston is connected with the gun, while the other is 8 inches in diameter, and its piston is connected with two rows of 26 couples of Belleville springs, d. The two cylinders communicate through a check valve.
When the gun is in battery, the liquid fills the chamber of the 4 inch pump, while the piston of the 8 inch one is at the end of its stroke. A recoil has the effect of driving in the 4 inch piston and forcing the liquid into the other chamber, whose piston compresses the springs. At the end of the recoil, the gunner has only to act upon the valve by means of a handwheel in order to bring the gun into battery as slowly as he desires, through the action of the springs.
For high aiming, the gun and the movable part of its carriage are capable of revolving around a strong pin, c, so placed that the axis of the piece always passes very near the center of the embrasure, thus permitting of giving the latter minimum dimensions. The chamber of the 8 inch pump is provided with projections that slide between circular guides, and carries the strap of a small hydraulic piston, p, that suffices to move the entire affair in a vertical plane, the gun and movable carriage being balanced by a counterpoise, q.
The projectiles are hoisted to the breech of the gun by a crane.
Between the outer armor and turret sufficient space is left for a man to enter, in order to make repairs when necessary.
Each of the rolled iron plates of which the turret consists weighs 19 tons. The cupolas that we have examined in this article have been constructed on the hypothesis than an enemy will not be able to bring into the field guns of much greater caliber than 6 inches.—Le Genie Civil.
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HIGH SPEED ON THE OCEAN.
To the Editor of the Scientific American:
Although not a naval engineer, I wish to reply to some arguments advanced by Capt. Giles, and published in the SCIENTIFIC AMERICAN of Jan. 2, 1886, in regard to high speed on the ocean.
Capt. Giles argues that because quadrupeds and birds do not in propelling themselves exert their force in a direct line with the plane of their motion, but at an angle to it, the same principle would, if applied to a steamship, increase its speed. But let us look at the subject from another standpoint. The quadruped has to support the weight of his body, and propel himself forward, with the same force. If the force be applied perpendicularly, the body is elevated, but not moved forward. If the force is applied horizontally, the body moves forward, but soon falls to the ground, because it is not supported. But when the force is applied at the proper angle, the body is moved forward and at the same time supported. Directly contrary to Capt. Giles' theory, the greater the speed of the quadruped, the nearer in a direct line with his motion does he apply the propulsive force, and vice versa. This may easily be seen by any one watching the motions of the horse, hound, deer, rabbit, etc., when in rapid motion. The water birds and animals, whose weight is supported by the water, do not exert the propulsive force in a downward direction, but in a direct line with the plane of their motion. The man who swims does not increase his motion by kicking out at an angle, but by drawing the feet together with the legs straight, thus using the water between them as a double inclined plane, on which his feet and legs slide and thus increase his motion. The weight of the steamship is already supported by the water, and all that is required of the propeller is to push her forward. If set so as to act in a direct line with the plane of motion, it will use all its force to push her forward; if set so as to use its force in a perpendicular direction, it will use all its force to raise her out of the water. If placed at an angle of 45 deg. with the plane of motion, half the force will be used in raising the ship out of the water, and only half will be left to push her forward.
ENOS M. RICKER.
Park Rapids, Minn., Jan. 23, 1886.
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SIBLEY COLLEGE LECTURES.
BY THE CORNELL UNIVERSITY NONRESIDENT LECTURERS IN MECHANICAL ENGINEERING.
PRINCIPLES AND METHODS OF BALANCING FORCES DEVELOPED IN MOVING BODIES.
BY CHAS. T. PORTER.
INTRODUCTION.
On appearing for the first time before this Association, which, as I am informed, comprises the faculty and the entire body of students of the Sibley College of Mechanical Engineering and the Mechanic Arts, a reminiscence of the founder of this College suggests itself to me, in the relation of which I beg first to be indulged.
In the years 184789 I lived in Rochester, N.Y., and formed a slight acquaintance with Mr. Sibley, whose home was then, as it has ever since been, in that city. Nearly twelve years afterward, in the summer of 1861, which will be remembered as the first year of our civil war, I met Mr. Sibley again. We happened to occupy a seat together in a car from New York to Albany. He recollected me, and we had a conversation which made a lasting impression on my memory. I said we had a conversation. That reminds me of a story told by my dear friend, of precious memory, Alexander L. Holley. One summer Mr. Holley accompanied a party of artists on an excursion to Mt. Katahdin, which, as you know, rises in almost solitary grandeur amid the forests and lakes of Maine. He wrote, in his inimitably happy style, an account of this excursion, which appeared some time after in Scribner's Monthly, elegantly illustrated with views of the scenery. Among other things, Mr. Holley related how he and Mr. Church painted the sketches for a grand picture of Mt. Katahdin. "That is," he explained, "Mr. Church painted, and I held the umbrella."
This describes the conversation which Mr. Sibley and I had. Mr. Sibley talked, and I listened. He was a good talker, and I flatter myself that I rather excel as a listener. On that occasion I did my best, for I knew whom I was listening to. I was listening to the man who combined bold and comprehensive grasp of thought, unerring foresight and sagacity, and energy of action and power of accomplishment, in a degree not surpassed, if it was equaled, among men.
Some years before, Mr. Sibley had created the Western Union Telegraph Company. At that time telegraphy was in a very depressed state. The country was to a considerable extent occupied by local lines, chartered under various State laws, and operated without concert. Four rival companies, organized under the Morse, the Bain, the House, and the Hughes patents, competed for the business. Telegraph stock was nearly valueless. Hiram Sibley, a man of the people, a resident of an inland city, of only moderate fortune, alone grasped the situation. He saw that the nature of the business, and the demands of the country, alike required that a single organization, in which all interests should be combined, should cover the entire land with its network, by means of which every center and every outlying point, distant as well as near, could communicate with each other directly, and that such an organization must be financially successful. He saw all this vividly, and realized it with the most intense earnestness of conviction. With Mr. Sibley, to be convinced was to act; and so he set about the task of carrying this vast scheme into execution. The result is well known. By his immense energy, the magnetic power with which he infused his own convictions into other minds, the direct, practical way in which he set about the work, and his indomitable perseverance, Mr. Sibley attained at last a phenomenal success.
But he was not then telling me anything about this. He was telling me of the construction of the telegraph line to the Pacific Coast. Here again Mr. Sibley had seen that which was hidden from others. This case differed from the former one in two important respects. Then Mr. Sibley had been dependent on the aid and cooperation of many persons; and this he had been able to secure. Now, he could not obtain help from a human being; but he had become able to act independently of any assistance.
He had made a careful study of the subject, in his thoroughly practical way, and had become convinced that such a line was feasible, and would be remunerative. At his instance a convention of telegraph men met in the city of New York, to consider the project. The feeling in this convention was extremely unfavorable to it. A committee reported against it unanimously, on three grounds—the country was destitute of timber, the line would be destroyed by the Indians, and if constructed and maintained, it would not pay expenses. Mr. Sibley found himself alone. An earnest appeal which he made from the report of the committee was received with derisive laughter. The idea of running a telegraph line through what was then a wilderness, roamed over for between one and two thousand miles of its breadth by bands of savages, who of course would destroy the line as soon as it was put up, and where repairs would be difficult and useless, even if the other objections to it were out of the way, struck the members of the convention as so exquisitely ludicrous that it seemed as if they would never be done laughing about it. If Mr. Sibley had advocated a line to the moon, they would hardly have seen in it greater evidence of lunacy. When he could be heard, he rose again and said: "Gentlemen, you may laugh, but if I was not so old, I would build the line myself." Upon this, of course, they laughed louder than ever. As they laughed, he grew mad, and shouted: "Gentlemen, I will bar the years, and do it." And he did it. Without help from any one, for every man who claimed a right to express an opinion upon it scouted the project as chimerical, and no capitalist would put a dollar in it, Hiram Sibley built the line of telegraph to San Francisco, risking in it all he had in the world. He set about the work with his customary energy, all obstacles vanished, and the line was completed in an incredibly short time. And from the day it was opened, it has proved probably the most profitable line of telegraph that has ever been constructed. There was the practicability, and there was the demand and the business to be done, and yet no living man could see it, or could be made to see it, except Hiram Sibley. "And today," he said, with honest pride, "today in New York, men to whom I went almost on my knees for help in building this line, and who would not give me a dollar, have solicited me to be allowed to buy stock in it at the rate of five dollars for one."
"But how about the Indians?" I asked. "Why," he replied, "we never had any trouble from the Indians. I knew we wouldn't have. Men who supposed I was such a fool as to go about this undertaking before that was all settled didn't know me. No Indian ever harmed that line. The Indians are the best friends we have got. You see, we taught the Indians the Great Spirit was in that line; and what was more, we proved it to them. It was, by all odds, the greatest medicine they ever saw. They fairly worshiped it. No Indian ever dared to do it harm."
"But," he added, "there was one thing I didn't count on. The border ruffians in Missouri are as bad as anybody ever feared the Indians might be. They have given us so much trouble that we are now building a line around that State, through Iowa and Nebraska. We are obliged to do it."
This opened another phase of the subject. The telegraph line to the Pacific had a value beyond that which could be expressed in money. It was perhaps the strongest of all the ties which bound California so securely to the Union, in the dark days of its struggle for existence. The secession element in Missouri recognized the importance of the line in this respect, and were persistent in their efforts to destroy it. We have seen by what means their purpose was thwarted.
I have always felt that, among the countless evidences of the ordering of Providence by which the war for the preservation of the Union was signalized, not the least striking was the raising up of this remarkable man, to accomplish alone, and in the very nick of time, a work which at once became of such national importance.
This is the man who has crowned his useful career, and shown again his eminently practical character and wise foresight, by the endowment of this College, which cannot fail to be a perennial source of benefit to the country whose interests he has done so much to promote, and which his remarkable sagacity and energy contributed so much to preserve.
We have an excellent rule, followed by all successful designers of machinery, which is, to make provision for the extreme case, for the most severe test to which, under normal conditions, and so far as practicable under abnormal conditions also, the machinery can be subjected. Then, of course, any demands upon it which are less than the extreme demand are not likely to give trouble. I shall apply this principle in addressing you today. In what I have to say, I shall speak directly to the youngest and least advanced minds among my auditors. If I am successful in making an exposition of my subject which shall be plain to them, then it is evident that I need not concern myself about being understood by the higher class men and the professors.
The subject to which your attention is now invited is
THE PRINCIPLES AND METHODS OF BALANCING FORCES DEVELOPED IN MOVING BODIES.
This is a subject with which every one who expects to be concerned with machinery, either as designer or constructor, ought to be familiar. The principles which underlie it are very simple, but in order to be of use, these need to be thoroughly understood. If they have once been mastered, made familiar, incorporated into your intellectual being, so as to be readily and naturally applied to every case as it arises, then you occupy a high vantage ground. In this particular, at least, you will not go about your work uncertainly, trying first this method and then that one, or leaving errors to be disclosed when too late to remedy them. On the contrary, you will make, first your calculations and then your plans, with the certainty that the result will be precisely what you intend.
Moreover, when you read discussions on any branch of this subject, you will not receive these into unprepared minds, just as apt to admit error as truth, and possessing no test by which to distinguish the one from the other; but you will be able to form intelligent judgments with respect to them. You will discover at once whether or not the writers are anchored to the sure holding ground of sound principles.
It is to be observed that I do not speak of balancing bodies, but of balancing forces. Forces are the realities with which, as mechanical engineers, you will have directly to deal, all through your lives. The present discussion is limited also to those forces which are developed in moving bodies, or by the motion of bodies. This limitation excludes the force of gravity, which acts on all bodies alike, whether at rest or in motion. It is, indeed, often desirable to neutralize the effect of gravity on machinery. The methods of doing this are, however, obvious, and I shall not further refer to them.
Two very different forces, or manifestations of force, are developed by the motion of bodies. These are
MOMENTUM AND CENTRIFUGAL FORCE.
The first of these forces is exerted by every moving body, whatever the nature of the path in which it is moving, and always in the direction of its motion. The latter force is exerted only by bodies whose path is a circle, or a curve of some form, about a central body or point, to which it is held, and this force is always at right angles with the direction of motion of the body.
Respecting momentum, I wish only to call your attention to a single fact, which will become of importance in the course of our discussion. Experiments on falling bodies, as well as all experience, show that the velocity of every moving body is the product of two factors, which must combine to produce it. Those factors are force and distance. In order to impart motion to the body, force must act through distance. These two factors may be combined in any proportions whatever. The velocity imparted to the body will vary as the square root of their product. Thus, in the case of any given body,
Let force 1, acting through distance 1, impart velocity 1. Then " 1, " " " 4, will " " 2, or " 2, " " " 2, " " " 2, or " 4, " " " 1, " " " 2; And " 1, " " " 9, " " " 3, or " 3, " " " 3, " " " 3, or " 9, " " " 1, " " " 3.
This table might be continued indefinitely. The product of the force into the distance will always vary as the square of the final velocity imparted. To arrest a given velocity, the same force, acting through the same distance, or the same product of force into distance, is required that was required to impart the velocity.
The fundamental truth which I now wish to impress upon your minds is that in order to impart velocity to a body, to develop the energy which is possessed by a body in motion, force must act through distance. Distance is a factor as essential as force. Infinite force could not impart to a body the least velocity, could not develop the least energy, without acting through distance.
This exposition of the nature of momentum is sufficient for my present purpose. I shall have occasion to apply it later on, and to describe the methods of balancing this force, in those cases in which it becomes necessary or desirable to do so. At present I will proceed to consider the second of the forces, or manifestations of force, which are developed in moving bodies—centrifugal force.
This force presents its claims to attention in all bodies which revolve about fixed centers, and sometimes these claims are presented with a good deal of urgency. At the same time, there is probably no subject, about which the ideas of men generally are more vague and confused. This confusion is directly due to the vague manner in which the subject of centrifugal force is treated, even by our best writers. As would then naturally be expected, the definitions of it commonly found in our handbooks are generally indefinite, or misleading, or even absolutely untrue.
Before we can intelligently consider the principles and methods of balancing this force, we must get a correct conception of the nature of the force itself. What, then, is centrifugal force? It is an extremely simple thing; a very ordinary amount of mechanical intelligence is sufficient to enable one to form a correct and clear idea of it. This fact renders it all the more surprising that such inaccurate and confused language should be employed in its definition. Respecting writers, also, who use language with precision, and who are profound masters of this subject, it must be said that, if it had been their purpose to shroud centrifugal force in mystery, they could hardly have accomplished this purpose more effectually than they have done, to minds by whom it was not already well understood.
Let us suppose a body to be moving in a circular path, around a center to which it is firmly held; and let us, moreover, suppose the impelling force, by which the body was put in motion, to have ceased; and, also, that the body encounters no resistance to its motion. It is then, by our supposition, moving in its circular path with a uniform velocity, neither accelerated nor retarded. Under these conditions, what is the force which is being exerted on this body? Clearly, there is only one such force, and that is, the force which holds it to the center, and compels it, in its uniform motion, to maintain a fixed distance from this center. This is what is termed centripetal force. It is obvious, that the centripetal force, which holds this revolving body to the center, is the only force which is being exerted upon it.
Where, then, is the centrifugal force? Why, the fact is, there is not any such thing. In the dynamical sense of the term "force," the sense in which this term is always understood in ordinary speech, as something tending to produce motion, and the direction of which determines the direction in which motion of a body must take place, there is, I repeat, no such thing as centrifugal force.
There is, however, another sense in which the term "force" is employed, which, in distinction from the above, is termed a statical sense. This "statical force" is the force by the exertion of which a body keeps still. It is the force of inertia—the resistance which all matter opposes to a dynamical force exerted to put it in motion. This is the sense in which the term "force" is employed in the expression "centrifugal force." Is that all? you ask. Yes; that is all.
I must explain to you how it is that a revolving body exerts this resistance to being put in motion, when all the while it is in motion, with, according to our above supposition, a uniform velocity. The first law of motion, so far as we now have occasion to employ it, is that a body, when put in motion, moves in a straight line. This a moving body always does, unless it is acted on by some force, other than its impelling force, which deflects it, or turns it aside, from its direct line of motion. A familiar example of this deflecting force is afforded by the force of gravity, as it acts on a projectile. The projectile, discharged at any angle of elevation, would move on in a straight line forever, but, first, it is constantly retarded by the resistance of the atmosphere, and, second, it is constantly drawn downward, or made to fall, by the attraction of the earth; and so instead of a straight line it describes a curve, known as the trajectory.
Now a revolving body, also, has the same tendency to move in a straight line. It would do so, if it were not continually deflected from this line. Another force is constantly exerted upon it, compelling it, at every successive point of its path, to leave the direct line of motion, and move on a line which is everywhere equally distant from the center to which it is held. If at any point the revolving body could get free, and sometimes it does get free, it would move straight on, in a line tangent to the circle at the point of its liberation. But if it cannot get free, it is compelled to leave each new tangential direction, as soon as it has taken it.
This is illustrated in the above figure. The body, A, is supposed to be revolving in the direction indicated by the arrow, in the circle, A B F G, around the center, O, to which it is held by the cord, O A. At the point, A, it is moving in the tangential direction, A D. It would continue to move in this direction, did not the cord, O A, compel it to move in the arc, A C. Should this cord break at the point, A, the body would move; straight on toward D, with whatever velocity it had.
You perceive now what centrifugal force is. This body is moving in the direction, A D. The centripetal force, exerted through the cord, O A, pulls it aside from this direction of motion. The body resists this deflection, and this resistance is its centrifugal force.
Centrifugal force is, then, properly defined to be the disposition of a revolving body to move in a straight line, and the resistance which such a body opposes to being drawn aside from a straight line of motion. The force which draws the revolving body continually to the center, or the deflecting force, is called the centripetal force, and, aside from the impelling and retarding forces which act in the direction of its motion, the centripetal force is, dynamically speaking, the only force which is exerted on the body.
It is true, the resistance of the body furnishes the measure of the centripetal force. That is, the centripetal force must be exerted in a degree sufficient to overcome this resistance, if the body is to move in the circular path. In this respect, however, this case does not differ from every other case of the exertion of force. Force is always exerted to overcome resistance: otherwise it could not be exerted. And the resistance always furnishes the exact measure of the force. I wish to make it entirely clear, that in the dynamical sense of the term "force," there is no such thing as centrifugal force. The dynamical force, that which produces motion, is the centripetal force, drawing the body continually from the tangential direction, toward the center; and what is termed centrifugal force is merely the resistance which the body opposes to this deflection, precisely like any other resistance to a force.
The centripetal force is exerted on the radial line, as on the line, A O, Fig. 1, at right angles with the direction in which the body is moving; and draws it directly toward the center. It is, therefore, necessary that the resistance to this force shall also be exerted on the same line, in the opposite direction, or directly from the center. But this resistance has not the least power or tendency to produce motion in the direction in which it is exerted, any more than any other resistance has.
We have been supposing a body to be firmly held to the center, so as to be compelled to revolve about it in a fixed path. But the bond which holds it to the center may be elastic, and in that case, if the centrifugal force is sufficient, the body will be drawn from the center, stretching the elastic bond. It may be asked if this does not show centrifugal force to be a force tending to produce motion from the center. This question is answered by describing the action which really takes place. The revolving body is now imperfectly deflected. The bond is not strong enough to compel it to leave its direct line of motion, and so it advances a certain distance along this tangential line. This advance brings the body into a larger circle, and by this enlargement of the circle, assuming the rate of revolution to be maintained, its centrifugal force is proportionately increased. The deflecting power exerted by the elastic bond is also increased by its elongation. If this increase of deflecting force is no greater than the increase of centrifugal force, then the body will continue on in its direct path; and when the limit of its elasticity is reached, the deflecting bond will be broken. If, however, the strength of the deflecting bond is increased by its elongation in a more rapid ratio than the centrifugal force is increased by the enlargement of the circle, then a point will be reached in which the centripetal force will be sufficient to compel the body to move again in the circular path.
Sometimes the centripetal force is weak, and opportunity is afforded to observe this action, and see its character exhibited. A common example of weak centripetal force is the adhesion of water to the face of a revolving grindstone. Here we see the deflecting force to become insufficient to compel the drops of water longer to leave their direct paths, and so these do not longer leave their direct paths, but move on in those paths, with the velocity they have at the instant of leaving the stone, flying off on tangential lines.
If, however, a fluid be poured on the side of the revolving wheel near the axis, it will move out to the rim on radial lines, as may be observed on car wheels universally. The radial lines of black oil on these wheels look very much as if centrifugal force actually did produce motion, or had at least a very decided tendency to produce motion, in the radial direction. This interesting action calls for explanation. In this action the oil moves outward gradually, or by inconceivably minute steps. Its adhesion being overcome in the least possible degree, it moves in the same degree tangentially. In so doing it comes in contact with a point of the surface which has a motion more rapid than its own. Its inertia has now to be overcome, in the same degree in which it had overcome the adhesion. Motion in the radial direction is the result of these two actions, namely, leaving the first point of contact tangentially and receiving an acceleration of its motion, so that this shall be equal to that of the second point of contact. When we think about the matter a little closely, we see that at the rim of the wheel the oil has perhaps ten times the velocity of revolution which it had on leaving the journal, and that the mystery to be explained really is, How did it get that velocity, moving out on a radial line? Why was it not left behind at the very first? Solely by reason of its forward tangential motion. That is the answer.
When writers who understand the subject talk about the centripetal and centrifugal forces being different names for the same force, and about equal action and reaction, and employ other confusing expressions, just remember that all they really mean is to express the universal relation between force and resistance. The expression "centrifugal force" is itself so misleading, that it becomes especially important that the real nature of this socalled force, or the sense in which the term "force" is used in this expression, should be fully explained.[1] This force is now seen to be merely the tendency of a revolving body to move in a straight line, and the resistance which it opposes to being drawn aside from that line. Simple enough! But when we come to consider this action carefully, it is wonderful how much we find to be contained in what appears so simple. Let us see.
[Footnote 1: I was led to study this subject in looking to see what had become of my first permanent investment, a small venture, made about thirtyfive years ago, in the "Sawyer and Gwynne static pressure engine." This was the highsounding name of the Keely motor of that day, an imposition made possible by the confused ideas prevalent on this very subject of centrifugal force.]
FIRST.—I have called your attention to the fact that the direction in which the revolving body is deflected from the tangential line of motion is toward the center, on the radial line, which forms a right angle with the tangent on which the body is moving. The first question that presents itself is this: What is the measure or amount of this deflection? The answer is, this measure or amount is the versed sine of the angle through which the body moves.
Now, I suspect that some of you—some of those whom I am directly addressing—may not know what the versed sine of an angle is; so I must tell you. We will refer again to Fig. 1. In this figure, O A is one radius of the circle in which the body A is revolving. O C is another radius of this circle. These two radii include between them the angle A O C. This angle is subtended by the arc A C. If from the point O we let fall the line C E perpendicular to the radius O A, this line will divide the radius O A into two parts, O E and E A. Now we have the three interior lines, or the three lines within the circle, which are fundamental in trigonometry. C E is the sine, O E is the cosine, and E A is the versed sine of the angle A O C. Respecting these three lines there are many things to be observed. I will call your attention to the following only:
First.—Their length is always less than the radius. The radius is expressed by 1, or unity. So, these lines being less than unity, their length is always expressed by decimals, which mean equal to such a proportion of the radius.
Second.—The cosine and the versed sine are together equal to the radius, so that the versed sine is always 1, less the cosine.
Third.—If I diminish the angle A O C, by moving the radius O C toward O A, the sine C E diminishes rapidly, and the versed sine E A also diminishes, but more slowly, while the cosine O E increases. This you will see represented in the smaller angles shown in Fig. 2. If, finally, I make O C to coincide with O A, the angle is obliterated, the sine and the versed sine have both disappeared, and the cosine has become the radius.
Fourth.—If, on the contrary, I enlarge the angle A O C by moving the radius O C toward O B, then the sine and the versed sine both increase, and the cosine diminishes; and if, finally, I make O C coincide with O B, then the cosine has disappeared, the sine has become the radius O B, and the versed sine has become the radius O A, thus forming the two sides inclosing the right angle A O B. The study of this explanation will make you familiar with these important lines. The sine and the cosine I shall have occasion to employ in the latter part of my lecture. Now you know what the versed sine of an angle is, and are able to observe in Fig. 1 that the versed sine A E, of the angle A O C, represents in a general way the distance that the body A will be deflected from the tangent A D toward the center O while describing the arc A C.
The same law of deflection is shown, in smaller angles, in Fig. 2. In this figure, also, you observe in each of the angles A O B and A O C that the deflection, from the tangential direction toward the center, of a body moving in the arc A C is represented by the versed sine of the angle. The tangent to the arc at A, from which this deflection is measured, is omitted in this figure to avoid confusion. It is shown sufficiently in Fig. 1. The angles in Fig. 2 are still pretty large angles, being 12 deg. and 24 deg. respectively. These large angles are used for convenience of illustration; but it should be explained that this law does not really hold in them, as is evident, because the arc is longer than the tangent to which it would be connected by a line parallel with the versed sine. The law is absolutely true only when the tangent and arc coincide, and approximately so for exceedingly small angles.
In reality, however, we have only to do with the case in which the arc and the tangent do coincide, and in which the law that the deflection is equal to the versed sine of the angle is absolutely true. Here, in observing this most familiar thing, we are, at a single step, taken to that which is utterly beyond our comprehension. The angles we have to consider disappear, not only from our sight, but even from our conception. As in every other case when we push a physical investigation to its limit, so here also, we find our power of thought transcended, and ourselves in the presence of the infinite.
We can discuss very small angles. We talk familiarly about the angle which is subtended by 1" of arc. On Fig. 2, a short line is drawn near to the radius O A'. The distance between O A' and this short line is 1 deg. of the arc A' B'. If we divide this distance by 3,600, we get 1" of arc. The upper line of the Table of versed sines given below is the versed sine of 1" of arc. It takes 1,296,000 of these angles to fill a circular space. These are a great many angles, but they do not make a circle. They make a polygon. If the radius of the circumscribed circle of this polygon is 1,296,000 feet, which is nearly 213 geographical miles, each one of its sides will be a straight line, 6.283 feet long. On the surface of the earth, at the equator, each side of this polygon would be onesixtieth of a geographical mile, or 101.46 feet. On the orbit of the moon, at its mean distance from the earth, each of these straight sides would be about 6,000 feet long.
The best we are able to do is to conceive of a polygon having an infinite number of sides, and so an infinite number of angles, the versed sines of which are infinitely small, and having, also, an infinite number of tangential directions, in which the body can successively move. Still, we have not reached the circle. We never can reach the circle. When you swing a sling around your head, and feel the uniform stress exerted on your hand through the cord, you are made aware of an action which is entirely beyond the grasp of our minds and the reach of our analysis.
So always in practical operation that law is absolutely true which we observe to be approximated to more and more nearly as we consider smaller and smaller angles, that the versed sine of the angle is the measure of its deflection from the straight line of motion, or the measure of its fall toward the center, which takes place at every point in the motion of a revolving body.
Then, assuming the absolute truth of this law of deflection, we find ourselves able to explain all the phenomena of centrifugal force, and to compute its amount correctly in all cases.
We have now advanced two steps. We have learned the direction and the measure of the deflection, which a revolving body continually suffers, and its resistance to which is termed centrifugal force. The direction is toward the center, and the measure is the versed sine of the angle.
SECOND.—We next come to consider what are known as the laws of centrifugal force. These laws are four in number. They are, that the amount of centrifugal force exerted by a revolving body varies in four ways.
First.—Directly as the weight of the body.
Second.—In a given circle of revolution, as the square of the speed or of the number of revolutions per minute; which two expressions in this case mean the same thing.
Third.—With a given number of revolutions per minute, or a given angular velocity[1] directly as the radius of the circle; and
Fourth.—With a given actual velocity, or speed in feet per minute, inversely as the radius of the circle.
[Footnote 1: A revolving body is said to have the same angular velocity, when it sweeps through equal angles in equal times. Its actual velocity varies directly as the radius of the circle in which it is revolving.]
Of course there is a reason for these laws. You are not to learn them by rote, or to accept them on any authority. You are taught not to accept any rule or formula on authority, but to demand the reason for it—to give yourselves no rest until you know the why and wherefore, and comprehend these fully. This is education, not cramming the mind with mere facts and rules to be memorized, but drawing out the mental powers into activity, strengthening them by use and exercise, and forming the habit, and at the same time developing the power, of penetrating to the reason of things.
In this way only, you will be able to meet the requirement of a great educator, who said: "I do not care to be told what a young man knows, but what he can do." I wish here to add my grain to the weight of instruction which you receive, line upon line, precept on precept, on this subject.
The reason for these laws of centrifugal force is an extremely simple one. The first law, that this force varies directly as the weight of the body, is of course obvious. We need not refer to this law any further. The second, third, and fourth laws merely express the relative rates at which a revolving body is deflected from the tangential direction of motion, in each of the three cases described, and which cases embrace all possible conditions.
These three rates of deflection are exhibited in Fig. 2. An examination of this figure will give you a clear understanding of them. Let us first suppose a body to be revolving about the point, O, as a center, in a circle of which A B C is an arc, and with a velocity which will carry it from A to B in one second of time. Then in this time the body is deflected from the tangential direction a distance equal to A D, the versed sine of the angle A O B. Now let us suppose the velocity of this body to be doubled in the same circle. In one second of time it moves from A to C, and is deflected from the tangential direction of motion a distance equal to A E, the versed sine of the angle, A O C. But A E is four times A D. Here we see in a given circle of revolution the deflection varying as the square of the speed. The slight error already pointed out in these large angles is disregarded.
The following table will show, by comparison of the versed sines of very small angles, the deflection in a given circle varying as the square of the speed, when we penetrate to them, so nearly that the error is not disclosed at the fifteenth place of decimals.
The versed sine of 1" is 0.000,000,000,011,752 " " " " 2" is 0.000,000,000,047,008 " " " " 3" is 0.000,000,000,105,768 " " " " 4" is 0.000,000,000,188,032 " " " " 5" is 0.000,000,000,293,805 " " " " 6" is 0.000,000,000,423,072 " " " " 7" is 0.000,000,000,575,848 " " " " 8" is 0.000,000,000,752,128 " " " " 9" is 0.000,000,000,951,912 " " " " 10" is 0.000,000,001,175,222 " " " " 100" is 0.000,000,117,522,250
You observe the deflection for 10" of arc is 100 times as great, and for 100" of arc is 10,000 times as great as it is for 1" of arc. So far as is shown by the 15th place of decimals, the versed sine varies as the square of the angle; or, in a given circle, the deflection, and so the centrifugal force, of a revolving body varies as the square of the speed.
The reason for the third law is equally apparent on inspection of Fig. 2. It is obvious, that in the case of bodies making the same number of revolutions in different circles, the deflection must vary directly as the diameter of the circle, because for any given angle the versed sine varies directly as the radius. Thus radius O A' is twice radius O A, and so the versed sine of the arc A' B' is twice the versed sine of the arc A B. Here, while the angular velocity is the same, the actual velocity is doubled by increase in the diameter of the circle, and so the deflection is doubled. This exhibits the general law, that with a given angular velocity the centrifugal force varies directly as the radius or diameter of the circle.
We come now to the reason for the fourth law, that, with a given actual velocity, the centrifugal force varies inversely as the diameter of the circle. If any of you ever revolved a weight at the end of a cord with some velocity, and let the cord wind up, suppose around your hand, without doing anything to accelerate the motion, then, while the circle of revolution was growing smaller, the actual velocity continuing nearly uniform, you have felt the continually increasing stress, and have observed the increasing angular velocity, the two obviously increasing in the same ratio. That is the operation or action which the fourth law of centrifugal force expresses. An examination of this same figure (Fig. 2) will show you at once the reason for it in the increasing deflection which the body suffers, as its circle of revolution is contracted. If we take the velocity A' B', double the velocity A B, and transfer it to the smaller circle, we have the velocity A C. But the deflection has been increasing as we have reduced the circle, and now with one half the radius it is twice as great. It has increased in the same ratio in which the angular velocity has increased. Thus we see the simple and necessary nature of these laws. They merely express the different rates of deflection of a revolving body in these different cases.
THIRD.—We have a coefficient of centrifugal force, by which we are enabled to compute the amount of this resistance of a revolving body to deflection from a direct line of motion in all cases. This is that coefficient. The centrifugal force of a body making one revolution per minute, in a circle of one foot radius, is 0.000341 of the weight of the body.
According to the above laws, we have only to multiply this coefficient by the square of the number of revolutions made by the body per minute, and this product by the radius of the circle in feet, or in decimals of a foot, and we have the centrifugal force, in terms of the weight of the body. Multiplying this by the weight of the body in pounds, we have the centrifugal force in pounds.
Of course you want to know how this coefficient has been found out, and how you can be sure it is correct. I will tell you a very simple way. There are also mathematical methods of ascertaining this coefficient, which your professors, if you ask them, will let you dig out for yourselves. The way I am going to tell you I found out for myself, and that, I assure you, is the only way to learn anything, so that it will stick; and the more trouble the search gives you, the darker the way seems, and the greater the degree of perseverance that is demanded, the more you will appreciate the truth when you have found it, and the more complete and permanent your possession of it will be.
The explanation of this method may be a little more abstruse than the explanations already given, but it is very simple and elegant when you see it, and I fancy I can make it quite clear. I shall have to preface it by the explanation of two simple laws. The first of these is, that a body acted on by a constant force, so as to have its motion uniformly accelerated, suppose in a straight line, moves through distances which increase as the square of the time that the accelerating force continues to be exerted.
The necessary nature of this law, or rather the action of which this law is the expression, is shown in Fig. 3.
Let the distances A B, B C, C D, and D E in this figure represent four successive seconds of time. They may just as well be conceived to represent any other equal units, however small. Seconds are taken only for convenience. At the commencement of the first second, let a body start from a state of rest at A, under the action of a constant force, sufficient to move it in one second through a distance of one foot. This distance also is taken only for convenience. At the end of this second, the body will have acquired a velocity of two feet per second. This is obvious because, in order to move through one foot in this second, the body must have had during the second an average velocity of one foot per second. But at the commencement of the second it had no velocity. Its motion increased uniformly. Therefore, at the termination of the second its velocity must have reached two feet per second. Let the triangle A B F represent this accelerated motion, and the distance, of one foot, moved through during the first second, and let the line B F represent the velocity of two feet per second, acquired by the body at the end of it. Now let us imagine the action of the accelerating force suddenly to cease, and the body to move on merely with the velocity it has acquired. During the next second it will move through two feet, as represented by the square B F C I. But in fact, the action of the accelerating force does not cease. This force continues to be exerted, and produces on the body during the next second the same effect that it did during the first second, causing it to move through an additional foot of distance, represented by the triangle F I G, and to have its velocity accelerated two additional feet per second, as represented by the line I G. So in two seconds the body has moved through four feet. We may follow the operation of this law as far as we choose. The figure shows it during four seconds, or any other unit, of time, and also for any unit of distance. Thus:
Time 1 Distance 1 " 2 " 4 " 3 " 9 " 4 " 16
So it is obvious that the distance moved through by a body whose motion is uniformly accelerated increases as the square of the time.
But, you are asking, what has all this to do with a revolving body? As soon as your minds can be started from a state of rest, you will perceive that it has everything to do with a revolving body. The centripetal force, which acts upon a revolving body to draw it to the center, is a constant force, and under it the revolving body must move or be deflected through distances which increase as the squares of the times, just as any body must do when acted on by a constant force. To prove that a revolving body obeys this law, I have only to draw your attention to Fig. 2. Let the equal arcs, A B and B C, in this figure represent now equal times, as they will do in case of a body revolving in this circle with a uniform velocity. The versed sines of the angles, A O B and A O C, show that in the time, A C, the revolving body was deflected four times as far from the tangent to the circle at A as it was in the time, A B. So the deflection increased as the square of the time. If on the table already given, we take the seconds of arc to represent equal times, we see the versed sine, or the amount of deflection of a revolving body, to increase, in these minute angles, absolutely so far as appears up to the fifteenth place of decimals, as the square of the time.
The standard from which all computations are made of the distances passed through in given times by bodies whose motion is uniformly accelerated, and from which the velocity acquired is computed when the accelerating force is known, and the force is found when the velocity acquired or the rate of acceleration is known, is the velocity of a body falling to the earth. It has been established by experiment, that in this latitude near the level of the sea, a falling body in one second falls through a distance of 16.083 feet, and acquires a velocity of 32.166 feet per second; or, rather, that it would do so if it did not meet the resistance of the atmosphere. In the case of a falling body, its weight furnishes, first, the inertia, or the resistance to motion, that has to be overcome, and affords the measure of this resistance, and, second, it furnishes the measure of the attraction of the earth, or the force exerted to overcome its resistance. Here, as in all possible cases, the force and the resistance are identical with each other. The above is, therefore, found in this way to be the rate at which the motion of any body will be accelerated when it is acted on by a constant force equal to its weight, and encounters no resistance.
It follows that a revolving body, when moving uniformly in any circle at a speed at which its deflection from a straight line of motion is such that in one second this would amount to 16.083 feet, requires the exertion of a centripetal force equal to its weight to produce such deflection. The deflection varying as the square of the time, in 0.01 of a second this deflection will be through a distance of 0.0016083 of a foot.
Now, at what speed must a body revolve, in a circle of one foot radius, in order that in 0.01 of one second of time its deflection from a tangential direction shall be 0.0016083 of a foot? This decimal is the versed sine of the arc of 3 deg.15', or of 3.25 deg.. This angle is so small that the departure from the law that the deflection is equal to the versed sine of the angle is too slight to appear in our computation. Therefore, the arc of 3.25 deg. is the arc of a circle of one foot radius through which a body must revolve in 0.01 of a second of time, in order that the centripetal force, and so the centrifugal force, shall be equal to its weight. At this rate of revolution, in one second the body will revolve through 325 deg., which is at the rate of 54.166 revolutions per minute.
Now there remains only one question more to be answered. If at 54.166 revolutions per minute the centrifugal force of a body is equal to its weight, what will its centrifugal force be at one revolution per minute in the same circle?
To answer this question we have to employ the other extremely simple law, which I said I must explain to you. It is this: The acceleration and the force vary in a constant ratio with each other. Thus, let force 1 produce acceleration 1, then force 1 applied again will produce acceleration 1 again, or, in other words, force 2 will produce acceleration 2, and so on. This being so, and the amount of the deflection varying as the squares of the speeds in the two cases, the centrifugal force of a body making one revolution per minute in a circle of
1 squared one foot radius will be ————— = 0.000341 54.166 squared
—the coefficient of centrifugal force.
There is another mode of making this computation, which is rather neater and more expeditious than the above. A body making one revolution per minute in a circle of one foot radius will in one second revolve through an arc of 6 deg.. The versed sine of this arc of 6 deg. is 0.0054781046 of a foot. This is, therefore, the distance through which a body revolving at this rate will be deflected in one second. If it were acted on by a force equal to its weight, it would be deflected through the distance of 16.083 feet in the same time. What is the deflecting force actually exerted upon it? Of
0.0054781046 course, it is ——————. 16.083
This division gives 0.000341 of its weight as such deflecting force, the same as before.
In taking the versed sine of 6 deg., a minute error is involved, though not one large enough to change the last figure in the above quotient. The law of uniform acceleration does not quite hold when we come to an angle so large as 6 deg.. If closer accuracy is demanded, we can attain it, by taking the versed sine for 1 deg., and multiplying this by 6 squared. This gives as a product 0.0054829728, which is a little larger than the versed sine of 6 deg..
I hope I have now kept my promise, and made it clear how the coefficient of centrifugal force may be found in this simple way.
We have now learned several things about centrifugal force. Let me recapitulate. We have learned:
1st. The real nature of centrifugal force. That in the dynamical sense of the term force, this is not a force at all: that it is not capable of producing motion, that the force which is really exerted on a revolving body is the centripetal force, and what we are taught to call centrifugal force is nothing but the resistance which a revolving body opposes to this force, precisely like any other resistance.
2d. The direction of the deflection, to which the centrifugal force is the resistance, which is straight to the center.
3d. The measure of this deflection; the versed sine of the angle.
4th. The reason of the laws of centrifugal force; that these laws merely express the relative amount of the deflection, and so the amount of the force required to produce the deflection, and of the resistance of the revolving body to it, in all different cases.
5th. That the deflection of a revolving body presents a case analogous to that of uniformly accelerated motion, under the action of a constant force, similar to that which is presented by falling bodies;[1] and finally,
6th. How to find the coefficient, by which the amount of centrifugal force exerted in any case may be computed.
[Footnote 1: A body revolving with a uniform velocity in a horizontal plane would present the only case of uniformly accelerated motion that is possible to be realized under actual conditions.]
I now pass to some other features.
First.—You will observe that, relatively to the center, a revolving body, at any point in its revolution, is at rest. That is, it has no motion, either from or toward the center, except that which is produced by the action of the centripetal force. It has, therefore, this identity also with a falling body, that it starts from a state of rest. This brings us to a far more comprehensive definition of centrifugal force. This is the resistance which a body opposes to being put in motion, at any velocity acquired in any time, from a state of rest. Thus centrifugal force reveals to us the measure of the inertia of matter. This inertia may be demonstrated and exhibited by means of apparatus constructed on this principle quite as accurately as it can be in any other way.
Second.—You will also observe the fact, that motion must be imparted to a body gradually. As distance, through which force can act, is necessary to the impartation of velocity, so also time, during which force can act, is necessary to the same result. We do not know how motion from a state of rest begins, any more than we know how a polygon becomes a circle. But we do know that infinite force cannot impart absolutely instantaneous motion to even the smallest body, or to a body capable of opposing the least resistance. Time being an essential element or factor in the impartation of velocity, if this factor be omitted, the least resistance becomes infinite.
We have a practical illustration of this truth in the explosion of nitroglycerine. If a small portion of this compound be exploded on the surface of a granite bowlder, in the open air, the bowlder will be rent into fragments. The explanation of this phenomenon common among the laborers who are the most numerous witnesses of it, which you have doubtless often heard, and which is accepted by ignorant minds without further thought, is that the action of nitroglycerine is downward. We know that such an idea is absurd.
The explosive force must be exerted in all directions equally. The real explanation is, that the explosive action of nitroglycerine is so nearly instantaneous, that the resistance of the atmosphere is very nearly equal to that of the rock; at any rate, is sufficient to cause the rock to be broken up. The rock yields to the force very nearly as readily as the atmosphere does.
Third. An interesting solution is presented here of what is to many an astronomical puzzle. When I was younger than I am now, I was greatly troubled to understand how it could be that if the moon was always falling to the earth, as the astronomers assured us it was, it should never reach it, nor have its falling velocity accelerated. In popular treatises on astronomy, such for example as that of Professor Newcomb, this is explained by a diagram in which the tangential line is carried out as in Fig. 1, and by showing that in falling from the point A to the earth as a center, through distances increasing as the square of the time, the moon, having the tangential velocity that it has, could never get nearer to the earth than the circle in which it revolves around it. This is all very true, and very unsatisfactory. We know that this long tangential line has nothing to do with the motion of the moon, and while we are compelled to assent to the demonstration, we want something better. To my mind the better and more satisfactory explanation is found in the fact that the moon is forever commencing to fall, and is continually beginning to fall in a new direction. A revolving body, as we have seen, never gets past that point, which is entirely beyond our sight and our comprehension, of beginning to fall, before the direction of its fall is changed. So, under the attraction of the earth, the moon is forever leaving a new tangential direction of motion at the same rate, without acceleration.
(To be continued.)
* * * * *
COMPRESSED AIR POWER SCHEMES.
By J. STURGEON, Engineer of the Birmingham Compressed Air Power Company.
In the article on "Gas, Air, and Water Power" in the Journal for Dec. 8 last, you state that you await with some curiosity my reply to certain points in reference to the compressed air power schemes alluded to in that article. I now, therefore, take the liberty of submitting to you the arguments on my side of the question (which are substantially the same as those I am submitting to Mr. Hewson, the Borough Engineer of Leeds). The details and estimates for the Leeds scheme are not yet in a forward enough state to enable me to give them at present; but the whole case is sufficiently worked out for Birmingham to enable a fair deduction to be made therefrom as regards the utility of the system in other towns. In Birmingham, progress has been delayed owing to difficulties in procuring a site for the works, and other matters of detail. We have, however, recently succeeded in obtaining a suitable place, and making arrangements for railway siding, water supply, etc.; and we hope to be in a position to start early in the present year.
I inclose (1) a tabulated summary of the estimates for Birmingham divided into stages of 3,000 gross indicated horse power at a time; (2) a statement showing the cost to consumers in terms of indicated horse power and in different modes, more or less economical, of applying the air power in the consumers' engines; (3) a tracing showing the method of laying the mains; (4) a tracing showing the method of collecting the meter records at the central station, by means of electric apparatus, and ascertaining the exact amount of leakage. A short description of the two latter would be as well.
TABLE I.—Showing the Progressive Development of the Compressed Air System in stages of 3000 Indicated Horse Power (gross) at a Time, and the Profits at each Stage
Gross 3000 6000 9000 12,000 15,000 Indicated Ind. Ind. Ind. Ind. Ind. Horse Power H.P. H.P. H.P. H.P. H.P. at Central Works: 
Thousands of 1,080,000 2,160,000 3,240,000 4,320,000 5,400,000 Cubic Feet at 45 lbs. pressure at engines Deduction for 17,928 70,927 154,429 267,529 409,346 friction and leakage Estimated net 1,062,072 2,089,073 3,085,571 4,052,471 4,990,654 delivery 
CAPITAL EXPENDITURE Purchase and pre L12,500 (amounts below apply to extension of works) paration of land Machinery 27,854 L25,595 L25,595 L25,595 L25,595 Mains 10,328 10.328 10,328 10,328 10,328 Buildings 8,505 4,516 4,632 4,614 4,594 Parlimentary and general expenses, 20,000 .. .. .. .. royalty, &c. Engineering 3,268 1,820 1,825 1,824 8,823 Previous Capit 82,455 124,714 167,094 209,455 al Expenditure .. Total Cap. Exp. L82,455 L124,714 L167,094 L209,455 L251,795 
ANNUAL CHARGES Salaries, wages, & general working L6,405 L7,855 L9,305 L10,955 L12,480 expenses Repairs, renewals 2,780 5,198 7,622 10,045 12,467 &c.(reserve fund) Coal, water, &c. 1,950 3,900 5,850 7,800 9,750 Rates 370 674 980 1,285 1,585 Contingencies of horse power = 5 575 881 1,187 1,504 1,814 per cent on above Total Ann. Exp. L12,080 L18,508 L24,944 L31,589 L38,096 
Revenue at 5d. per 1000 cub. ft. 22,126 43,522 64,282 84,426 103,971 (average) Profit 12.18 p.ct. 20.06 p.ct. 23.54 p.ct. 25.22 p.ct. 26.16 p.ct. = 10,046 = 25,014 = 39,338 = 52,837 = 65,875 
TABLE II.—Cost of Air Power in Terms of Indicated Horse Power.
Abbreviated column headings:
Qty. Air: Quantity of Air at 45 lbs. Pressure required per Ind. H.P. per Hour.
Cost/Hr.: Cost per Hour at 5d. per 1000 Cubic Feet.
Cost/Hr. w/rebate: Cost per Hour with Rebate when Profits reach 26 per Cent.
Cost/Yr.: Cost per Annum (2700 Hours) at 5d. per 1000 Cubic Feet.
Cost/Yr. w/rebate: Cost per Annum with Rebate when Profits reach 26 per Cent.
Abbreviated row headings:
CASE 1.—Where air at 45 lbs. pressure is reheated to 320 deg. Fahr., and expanded to atmospheric pressure.
CASE 2.—Where air at 45 lbs. pressure is heated by boiling water to 212 deg. Fahr., and expanded to atmospheric pressure.
CASE 3.—Where air is used expansively without reheating, whereby intensely cold air is exhausted, and may be used for ice making, &c.
CASE 4.—Where air is heated to 212 deg. Fahr., and the terminal pressure is 11.3 lbs. above that of the atmosphere
CASE 5.—Where the air is used without heating, and cut off at onethird of the stroke, as in ordinary slidevalve engines
CASE 6.—Where the air is used without reheating and without expansion.
Qty. Air Cost/Hr. Cost/Hr. Cost/Yr. Cost/Yr. w/rebate w/rebate Cub. Ft. d. d. L s. d. L s. d.  CASE 1 125.4 0.627 0.596 7 1 1 6 14 01/2 CASE 2 140.4 0.702 0.667 7 17 11 7 10 0 CASE 3 178.2 0.891 0.847 10 0 51/2 9 10 51/2 CASE 4 170.2 0.851 0.809 9 11 51/2 9 1 101/2 CASE 5 258.0 1.290 1.226 14 10 3 13 15 9 CASE 6 331.8 1.659 1.576 18 13 3 17 14 7
The great thing to guard against is leakage. If the pipes were simply buried in the ground, it would be almost impossible to trace leakage, or even to know of its existence. The income of the company might be wasting away, and the loss never suspected until the quarterly returns from the meters were obtained from the inspectors. Only then would it be discovered that there must be a great leak (or it might be several leaks) somewhere. But how would it be possible to trace them among 20 or 30 miles of buried pipes? We cannot break up the public streets. The very existence of the concern depends upon (1) the daily checking of the meter returns, and comparison with the output from the air compressors, so as to ascertain the amount of leakage; (2) facility for tracing the locality of a leak; and (3) easy access to the mains with the minimum of disturbance to the streets. It will be readily understood, from the drawings, how this is effected. First, the pipes are laid in concrete troughs, near the surface of the road, with removable concrete covers strong enough to stand any overhead traffic. At intervals there are junctions for service connections, with street boxes and covers serving as inspection chambers. These chambers are also provided over the ballvalves, which serve as stopvalves in case of necessity, and are so arranged that in case of a serious breach in the portion of main between any two of them, the rush of air to the breach will blow them up to the corresponding seats and block off the broken portion of main. The air space around the pipe in the concrete trough will convey for a long distance the whistling noise of a leak; and the inspectors, by listening at the inspection openings, will thus be enabled to rapidly trace their way almost to the exact spot where there is an escape. They have then only to remove the top surface of road metal and the concrete cover in order to expose the pipe and get at the breach. Leaks would mostly be found at joints; and, by measuring from the nearest street opening, the inspectors would know where to break open the road to arrive at the probable locality of the leak. A very slight leak can be heard a long way off by its peculiar whistling sound.
The next point is to obtain a daily report of the condition of the mains and the amount of leakage. It would be impracticable to employ an army of meter inspectors to take the records daily from all the meters in the district. We therefore adopt the method of electric signaling shown in the second drawing. In the engineer's office, at the central station, is fixed the dial shown in Fig. 1. Each consumer's meter is fitted with the contactmaking apparatus shown in Pig. 4, and in an enlarged form in Figs. 5 and 6, by which a current is sent round the electromagnet, D (Fig. 1), attracting the armature, and drawing the disk forward sufficiently for the roller at I to pass over the center of one of the pins, and so drop in between that and the next pin, thus completing the motion, and holding the disk steadily opposite the figure. This action takes place on any meter completing a unit of measurement of (say) 1,000 cubic feet, at which point the contact makers touch. But suppose one meter should be moving very slowly, and so retaining contact for some time, while other meters were working rapidly; the armature at D would then be held up to the magnet by the prolonged contact maintained by the slow moving meter, and so prevent the quick working meters from actuating it; and they would therefore pass the contact points without recording. A meter might also stop dead at the point of contact on shutting off the air, and so hold up the armature; thus preventing others from acting. To obviate this, we apply the disengaging apparatus shown at L (Fig. 4). The contact maker works on the center, m, having an armature on its opposite end. On contact being made, at the same time that the magnet, D, is operated, the one at L is also operated, attracting the armature, and throwing over the end of the contact maker, l, on to the nonconducting side of the pin on the disk. Thus the whole movement is rendered practically instantaneous, and the magnet at D is set at liberty for the next operation. A resistance can be interposed at L, if necessary, to regulate the period of the operation. The whole of the meters work the common dial shown in Fig. 1, on which the gross results only are recorded; and this is all we want to know in this way. The action is so rapid, owing to the use of the magnetic disengaging gear, that the chances of two or more meters making contact at the same moment are rendered extremely small. Should such a thing happen, it would not matter, as it is only approximate results that we require in this case; and the error, if any, would add to the apparent amount of leakage, and so be on the right side. Of course, the record of each consumer's meter would be taken by the inspector at the end of every quarter, in order to make out the bill; and the totals thus obtained would be checked by the gross results indicated by the main dial. In this way, by a comparison of these results, a coefficient would soon be arrived at, by which the daily recorded results could be corrected to an extremely accurate measurement. At the end of the working day, the engineer has merely to take down from the dial in his office the total record of air measured to the consumers, also the output of air from the compressors, which he ascertains by means of a continuous counter on the engines, and the difference between the two will represent the loss. If the loss is trifling, he will pass it over; if serious, he will send out his inspectors to trace it. Thus there could be no long continued leakage, misuse, or robbery of the air, without the company becoming aware of the fact, and so being enabled to take measures to stop or prevent it. The foregoing are absolutely essential adjuncts to any scheme of public motive power supply by compressed air, without which we should be working in the dark, and could never be sure whether the company were losing or making money. With them, we know where we are and what we are doing.
Referring to the estimates given in Table I., I may explain that the item of repairs and renewals covers 10 per cent. on boilers and gas producers, 5 per cent. on engines, 5 per cent. on buildings, and 5 per cent. on mains. Considering that the estimates include ample fitting shops, with the best and most suitable tools, and that the wages list includes a staff of men whose chief work would be to attend to repairs, etc., I think the above allowances ample. Each item also includes 5 per cent. for contingencies. 
